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COMPLETE INDEPENDENCE
pretation remain apart and independent, each subject to its
own laws and conditions.
'Now the actual investigations of the following pages exhibit
Logic, in its practical aspect, as a system of processes carried on
by the aid of symbols having a definite interpretation, and
subject to laws founded upon that interpretation alone. But at
the same time they exhibit those laws as identical in form with
the laws of the general symbols of Algebra, with this single
addition, viz., that the symbols of Logic are further subject to a
special law [a?2 = x in the algebra of logic, which can be inter-
preted, among other ways, as "the class of all those things
common to a class x and itself is merely the class of*], to which
the symbols of quantity, as such, are not subject.' (That is, in
common algebra, it is not true that every x is equal to its square,
whereas in the Boolean algebra of logic, this is true.)
This programme is carried out in detail in the book. Boole
reduced logic to an extremely easy and simple type of algebra.
'Reasoning^ upon appropriate material becomes in this algebra
a matter of elementary manipulations of formulae far simpler
than most of those handled in a second year of school algebra.
Thus logic itself was brought under the sway of mathematics.
Since Boole's pioneering work his great invention has been
modified, improved, generalized, and extended in many direc-
tions. To-day symbolic or mathematical logic is indispensable
in any serious attempt to understand the nature of mathematics
and the state of its foundations on which the whole colossal
superstructure rests. The intricacy and delicacy of the diffi-
culties explored by the symbolic reasoning would, it is safe to
say, defy human reason if only the old, pre-Boole methods of
verbal logical arguments were at our disposal The daring origin-
ality of Boole's whole project needs no signpost. It is a land-
mark in itself.
Since 1899, when Hilbert published his classic on the founda-
tions of geometry, much attention has been given to the postu-
iational formulation of the several branches of mathematics.
This movement goes back as far as Euclid, but for some strange
reason - possibly because the techniques invented by Des-
cartes, Newton, Leibniz, Euler, Gauss, and others gave mathe-
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